Theorem dvlt0 23768

Description: A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Hypotheses

Ref	Expression
dvgt0.a	|- ( ph -> A e. RR )
dvgt0.b	|- ( ph -> B e. RR )
dvgt0.f	|- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
dvlt0.d	|- ( ph -> ( RR _D F ) : ( A (,) B ) --> ( -oo (,) 0 ) )

Assertion

Ref	Expression
dvlt0	|- ( ph -> F Isom < , `' < ( ( A [,] B ) , ran F ) )

Proof of Theorem dvlt0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvgt0.a	. 2 |- ( ph -> A e. RR )
2		dvgt0.b	. 2 |- ( ph -> B e. RR )
3		dvgt0.f	. 2 |- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
4		dvlt0.d	. 2 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> ( -oo (,) 0 ) )
5		gtso 10119	. 2 |- `' < Or RR
6	1, 2, 3, 4	dvgt0lem1 23765	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( -oo (,) 0 ) )
7		eliooord 12233	. . . . . . . . 9 |- ( ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( -oo (,) 0 ) -> ( -oo < ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) /\ ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 ) )
8	6, 7	syl 17	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( -oo < ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) /\ ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 ) )
9	8	simprd 479	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 )
10		cncff 22696	. . . . . . . . . . . 12 |- ( F e. ( ( A [,] B ) -cn-> RR ) -> F : ( A [,] B ) --> RR )
11	3, 10	syl 17	. . . . . . . . . . 11 |- ( ph -> F : ( A [,] B ) --> RR )
12	11	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> F : ( A [,] B ) --> RR )
13		simplrr 801	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( A [,] B ) )
14	12, 13	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) e. RR )
15		simplrl 800	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( A [,] B ) )
16	12, 15	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. RR )
17	14, 16	resubcld 10458	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) e. RR )
18		0red 10041	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> 0 e. RR )
19		iccssre 12255	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
20	1, 2, 19	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( A [,] B ) C_ RR )
21	20	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ RR )
22	21, 13	sseldd 3604	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR )
23	21, 15	sseldd 3604	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR )
24	22, 23	resubcld 10458	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR )
25		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x < y )
26	23, 22	posdifd 10614	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x < y <-> 0 < ( y - x ) ) )
27	25, 26	mpbid 222	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> 0 < ( y - x ) )
28		ltdivmul 10898	. . . . . . . 8 |- ( ( ( ( F ` y ) - ( F ` x ) ) e. RR /\ 0 e. RR /\ ( ( y - x ) e. RR /\ 0 < ( y - x ) ) ) -> ( ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 <-> ( ( F ` y ) - ( F ` x ) ) < ( ( y - x ) x. 0 ) ) )
29	17, 18, 24, 27, 28	syl112anc 1330	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 <-> ( ( F ` y ) - ( F ` x ) ) < ( ( y - x ) x. 0 ) ) )
30	9, 29	mpbid 222	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) < ( ( y - x ) x. 0 ) )
31	24	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. CC )
32	31	mul01d 10235	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( y - x ) x. 0 ) = 0 )
33	30, 32	breqtrd 4679	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) < 0 )
34	14, 16, 18	ltsubaddd 10623	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) < 0 <-> ( F ` y ) < ( 0 + ( F ` x ) ) ) )
35	33, 34	mpbid 222	. . . 4 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) < ( 0 + ( F ` x ) ) )
36	16	recnd 10068	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. CC )
37	36	addid2d 10237	. . . 4 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( 0 + ( F ` x ) ) = ( F ` x ) )
38	35, 37	breqtrd 4679	. . 3 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) < ( F ` x ) )
39		fvex 6201	. . . 4 |- ( F ` x ) e. _V
40		fvex 6201	. . . 4 |- ( F ` y ) e. _V
41	39, 40	brcnv 5305	. . 3 |- ( ( F ` x ) `' < ( F ` y ) <-> ( F ` y ) < ( F ` x ) )
42	38, 41	sylibr 224	. 2 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) `' < ( F ` y ) )
43	1, 2, 3, 4, 5, 42	dvgt0lem2 23766	1 |- ( ph -> F Isom < , `' < ( ( A [,] B ) , ran F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   class class class wbr 4653   `'ccnv 5113   ran crn 5115   -->wf 5884   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   -oocmnf 10072   < clt 10074   - cmin 10266   / cdiv 10684   (,)cioo 12175   [,]cicc 12178   -cn->ccncf 22679   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvne0  23774